Uniformly balanced words with linear complexity and prescribed letter frequencies

We consider the following problem. Let us fix a finite alphabet A; for any given d-uple of letter frequencies, how to construct an infinite word u over the alphabet A satisfying the following conditions: u has linear complexity function, u is uniformly balanced, the letter frequencies in u are given by the given d-uple. This paper investigates a construction method for such words based on the use of mixed multidimensional continued fraction algorithms.Comment: In Proceedings WORDS 2011, arXiv:1108.341

On the Structure of Bispecial Sturmian Words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of \emph{all} Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167

Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincar\'e Algorithm

The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries to the largest if possible (Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the median and the median to the largest (the Poincar\'e step), and by performing when possible Arnoux-Rauzy steps in priority. After renormalization it provides a piecewise fractional map of the standard $2$-simplex. We study here the factor complexity of its associated symbolic dynamical system, defined as an $S$-adic system. It is made of infinite words generated by the composition of sequences of finitely many substitutions, together with some restrictions concerning the allowed sequences of substitutions expressed in terms of a regular language. Here, the substitutions are provided by the matrices of the linear version of the algorithm. We give an upper bound for the linear growth of the factor complexity. We then deduce the convergence of the associated algorithm by unique ergodicity.Comment: 36 pages, 16 figure